A019279 -id:A019279 - cp's OEIS frontend

A138817 Concatenation of final digit of n-th Mersenne prime A000668(n), final digit of n-th even superperfect number A061652(n) and final digit of n-th perfect number A000396(n).

326, 748, 166, 748, 166, 166, 748, 748, 166, 166, 748, 748, 166, 748, 748, 748, 166, 166, 166, 748, 166, 166, 166, 166, 166, 166, 166, 748, 748, 166, 748, 748, 166, 748, 166, 166, 166, 166, 166

Offset: 1

Omar E. Pol, Apr 01 2008

Also, concatenation of final digit of n-th Mersenne prime A000668(n), final digit of n-th superperfect number A019279(n) and final digit of n-th perfect number A000396(n), if there are no odd superperfect numbers.
Also, concatenation of n-th term of A080172, A138125(n) and A094540(n).
a(1)=326. For n>1 a(n) is equal to 166 or 748, only.

L. C. Noll, Mersenne Prime Digits and Names.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000396, A000668, A019279, A061652, A080172, A094540, A138125, A138817.

A038843 Unitary superperfect numbers: numbers n such that usigma(usigma(n)) = 2*n, where usigma(n) is the sum of unitary divisors of n (A034448).

Original entry on oeis.org

2, 9, 165, 238, 1640, 4320, 10250, 10824, 13500, 23760, 58500, 66912, 425880, 520128, 873180, 931392, 1899744, 2129400, 2253888, 3276000, 4580064, 4668300, 13722800, 15459840, 40360320, 201801600, 439021440, 3809332800, 15359485680, 794436968640, 1407035080704

Offset: 1

Yasutoshi Kohmoto

May be called (2,2)-unitary perfect numbers, analogous to (k,l)-perfect numbers.

Sitaramaiah and Subbarao found the first 22 terms. Also in the sequence is 12189313382400. - Amiram Eldar, Feb 27 2019

V. Sitaramaiah and M. V. Subbarao, On the equation sigma*(sigma*(n)) = 2n, Utilitas Mathematica, Vol. 53 (1998), pp. 101-124.
Eric Weisstein's World of Mathematics, Super Unitary Perfect Number.
Tomohiro Yamada, Unitary super perfect numbers, Mathematica Pannonica, Volume 19, No. 1, 2008, pp. 37-47; Preprint, arXiv:0802.4377 [math.NT], 2008. Proves that 9 and 165 are the only odd terms of this sequence.
Tomohiro Yamada, 2 and 9 are the only biunitary superperfect numbers, Annales Univ. Sci. Budapest., Sec. Comp., Volume 48 (2018). Mentions this sequence.

Cf. A034448, A019279.

Cf. A064012 (usigma(usigma(n)) = 3n).

usigma[n_] := Times @@ (Apply[ Power, FactorInteger[n], {1}] + 1); n = 1; A038843 = {}; While[n < 10^7, If[ usigma[ usigma[n] ] == 2n, Print[n]; AppendTo[ A038843, n] ]; n++]; A038843 (* Jean-François Alcover, Dec 07 2011 *)

{usigma(n,s=1,fac,i)= fac=factor(n); for(i=1,matsize(fac)[1], s=s*(1+fac[i,1]^fac[i,2]) ); return(s);}
for(n=1,10^7, if(usigma(usigma(n))==2*n, print1(n, ", ")))

Corrected by Jason Earls, Aug 25 2001
More terms from Jud McCranie, Oct 28 2001
Offset corrected and a(28) from Donovan Johnson, Jul 23 2012
Name edited and a(29) from Amiram Eldar, Feb 27 2019
a(30)-a(31) from Giovanni Resta, Mar 08 2019

A138842 Concatenation of initial and final digits of n-th even superperfect number A061652(n).

Original entry on oeis.org

22, 44, 16, 64, 46, 66, 24, 14, 16, 36, 84, 84, 36, 24, 54, 74, 26, 16, 96, 14, 26, 16, 16, 26, 26, 26, 46, 24, 24, 26, 34, 84, 66, 24, 46, 36, 66, 26, 46, 64, 14, 64, 16, 66, 14, 86, 16

Offset: 1

Omar E. Pol, Apr 02 2008

Also, concatenation of initial and final digits of n-th superperfect number A019279(n), if there are no odd superperfect numbers.

Also, concatenation of A138124(n) and A138125(n).

Cf. A019279, A061652, A073729, A138124, A138125, A138838, A138840, A138841, A138843, A138844.

More terms from Jinyuan Wang, Mar 14 2020

A256438 Numbers m such that sigma(sigma(m-1)) = 2*(m-1).

Original entry on oeis.org

3, 5, 17, 65, 4097, 65537, 262145, 1073741825, 1152921504606846977, 309485009821345068724781057, 81129638414606681695789005144065, 85070591730234615865843651857942052865

Offset: 1

Jaroslav Krizek, Mar 29 2015

Numbers k such that A051027(k-1) = 2*(k-1).
Conjecture: numbers of the form 2^k+1 such that sigma(2^k) = prime p.
Prime terms: 3, 5, 17, 65537, ...
Supersequence of A249759.

			17 is in the sequence because sigma(sigma(17-1)) = 32 = 2*(17-1).

Cf. A000203, A051027, A019279, A249759.

[n: n in[2..10000000] | SumOfDivisors(SumOfDivisors(n-1)) eq 2*(n-1)];

with(numtheory): A256438:=n->`if`(sigma(sigma(n-1)) = 2*(n-1), n, NULL): seq(A256438(n), n=2..10^5); # Wesley Ivan Hurt, Mar 30 2015

Select[Range@ 1000000, DivisorSigma[1, DivisorSigma[1, # - 1]] == 2 (# - 1) &] (* Michael De Vlieger, Mar 29 2015 *)

isok(m) = sigma(sigma(m-1)) == 2*(m-1); \\ Michel Marcus, Feb 09 2020

a(n) = A019279(n) + 1. - Michel Marcus, Feb 09 2020

A272930 a(n) is the least k such that sigma(sigma(k)) = n*k, where sigma(n) is the sum of the divisors of n, or 0 if no such k exists.

Original entry on oeis.org

1, 2, 8, 15

Offset: 1

Franklin T. Adams-Watters, May 11 2016

If a(5) is not zero, it exceeds 5*10^11 (see A098223). Likewise for a(17).
a(6) to a(16) are 42, 24, 60, 168, 480, 4404480, 2200380, 57120, 217728, 1058148, 7526400. a(18) is 39352320.
Is a(n) in fact nonzero for every positive n? - Franklin T. Adams-Watters, Jan 22 2019 [who previously conjectured that it is]
a(19) to a(26) are 312792480, 1505806848, 341543854080, 83825280, 13460388480, 8530704000, 58350015360, 284430182400. - Michel Marcus, May 18 2016
From Michel Marcus, May 18 2016; Jul 19 2016, Aug 23 2016, Sep 06 2016: (Start)
a(17) <= 336421458837032140800;
a(27) <= 4641476998878720;
a(28) <= 23479734980782080;
a(29) <= 4670834235654671884800;
a(30) <= 7526652811748265000960;
a(31) <= 45781120625942782080;
a(32) <= 242094947364010540800;
a(33) <= 216462850095065333760000;
a(34) <= 2366077977040955880819916800;
a(35) <= 8076837429313362044375040000;
a(36) <= 2634106558176405916291008921600;
a(37) <= 299500004890186577026355605378405509365760000000;
a(38) <= 45103591381041833364829469933568000. (End)

			sigma(8) = 15. sigma(15) = 24 = 3*8. Since this does not work for any value smaller than 8, a(3) = 8.

See the links in A019278. - _Altug Alkan_, May 31 2016 and May 18 2016

Cf. A000203 (sigma), A098223, A051027, A019279, A007539, A019278.

with(numtheory):
a:=proc(n) local k :
for k while sigma(sigma(k))<>n*k do od : k end: # Robert FERREOL, Apr 11 2018

Table[SelectFirst[Range[10^2], Nest[DivisorSigma[1, #] &, #, 2] == n # &], {n, 4}] (* Michael De Vlieger, May 11 2016, Version 10 *)

a(n)=my(r=1);while(sigma(sigma(r))!=n*r,r++);r \\ works only if a(n) is not zero.

A138125 Final digit of n-th even superperfect number A061652(n).

Original entry on oeis.org

2, 4, 6, 4, 6, 6, 4, 4, 6, 6, 4, 4, 6, 4, 4, 4, 6, 6, 6, 4, 6, 6, 6, 6, 6, 6, 6, 4, 4, 6, 4, 4, 6, 4, 6, 6, 6, 6, 6, 4, 4, 4, 6, 6, 4, 6, 6

Offset: 1

Omar E. Pol, Apr 01 2008, corrected Apr 03 2008

Also, final digit of n-th superperfect number A019279(n), if there are no odd superperfect numbers.

			a(5)=6 because the 5th even superperfect number A061652(5) is 4096 and the final digit of 4096 is 6.
a(34)=4 because the final digit of 34th Mersenne prime is 7. a(39)=6 because the final digit of 39th Mersenne prime is 1.
.............................................................
............... SHORT TABLE OF FINAL DIGITS .................
.............................................................
Final digit of ..... Final digit of Even ..... Final digit of
Mersenne prime ..... Superperfect number ..... Perfect number
A000668 ............ A061652 ................. A000396........
(3) ................ (2) ..................... (6) ........... (For n=1, only)
(7) ................ (4) ..................... (8) ...........
(1) ................ (6) ..................... (6) ...........

Landon Curt Noll, Mersenne Prime Digits.

Cf. A000030, A000396, A000668, A077648, A019279, A061652, A135613, A135617, A138125.

Mod[#,10]&/@(2^(MersennePrimeExponent[Range[47]]-1)) (* Harvey P. Dale, Feb 23 2023 *)

a(1)=2. For n>1, if final digit of n-th Mersenne prime A000668(n) is equal to 1 then a(n)=6, otherwise a(n)=4.

a(40)-a(47) from Jinyuan Wang, Mar 14 2020

A138816 Concatenation of initial digit of n-th Mersenne prime A000668(n), initial digit of n-th even superperfect number A061652(n) and initial digit of n-th perfect number A000396(n).

Original entry on oeis.org

326, 742, 314, 168, 843, 168, 521, 212, 212, 631, 181, 181, 632, 521, 155

Offset: 1

Omar E. Pol, Apr 01 2008

Also, concatenation of initial digit of n-th Mersenne prime A000668(n), initial digit of n-th superperfect number A019279(n) and initial digit of n-th perfect number A000396(n), if there are no odd superperfect numbers.

Also, concatenation of A135613(n), A138124(n) and A135617(n).

L. C. Noll, Mersenne Prime Digits and Names.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000396, A000668, A019279, A061652, A135613, A135617, A138124, A138817.

a(13)-a(15) from Robert Price, Jun 16 2019

A138818 Concatenation of initial digit of n-th even superperfect number A061652(n), initial digit of n-th Mersenne prime A000668(n) and initial digit of n-th perfect number A000396(n).

Original entry on oeis.org

236, 472, 134, 618, 483, 618, 251, 122, 122, 361, 811, 811

Offset: 1

Omar E. Pol, Apr 05 2008

Also, concatenation of initial digit of n-th superperfect number A019279(n), initial digit of n-th Mersenne prime A000668(n) and initial digit of n-th perfect number A000396(n), if there are no odd superperfect numbers.

Also, concatenation of A138124(n), A135613(n) and A135617(n).

L. C. Noll, Mersenne Prime Digits and Names.
J. M. Pedersen, Known Perfect Numbers.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000396, A000668, A019279, A061652, A135613, A135617, A138124, A138816, A138817, A138819, A138841, A138842, A138843.

A138819 Concatenation of final digit of n-th even superperfect number A061652(n), final digit of n-th Mersenne prime A000668(n) and final digit of n-th perfect number A000396(n).

Original entry on oeis.org

236, 478, 616, 478, 616, 616, 478, 478, 616, 616, 478, 478, 616, 478, 478, 478, 616, 616, 616, 478, 616, 616, 616, 616, 616, 616, 616, 478, 478, 616, 478, 478, 616, 478, 616, 616, 616, 616, 616

Offset: 1

Omar E. Pol, Apr 05 2008

Also, concatenation of final digit of n-th superperfect number A019279(n), final digit of n-th Mersenne prime A000668(n) and final digit of n-th perfect number A000396(n), if there are no odd superperfect numbers.
Also, concatenation of A138125(n), A080172(n) and A094540(n).
For n>1 a(n) is equal to 478 or 616, only.
Note that, for n>1: if the final digit of n-th Mersenne prime A000668(n) is 1 then the final digit of n-th even superperfect number is 6 and the final digit of n-th perfect number also is 6, otherwise the final digit of n-th even superperfect number is 4 and the final digit of n-th perfect number is 8 (see example).

			===================================================================
.................. SHORT TABLE OF FINAL DIGITS ...................
===================================================================
... Final digit of even ..... Final digit of ..... Final digit of
... superperfect number ..... Mersenne prime ..... perfect number
........ A061652 ............... A000668 ............. A000396
===================================================================
n = 1 ..... (2) ................... (3) .................. (6)
n > 1 ..... (4) ................... (7) .................. (8)
n > 1 ..... (6) ................... (1) .................. (6)

L. C. Noll, Mersenne Prime Digits and Names.
J. M. Pedersen, Known Perfect Numbers.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000396, A000668, A019279, A061652, A080172, A094540, A138125, A138816, A138817, A138818, A138841, A138842, A138843.

A138124 Initial digit of n-th even superperfect number A061652(n).

Original entry on oeis.org

2, 4, 1, 6, 4, 6, 2, 1, 1, 3, 8, 8, 3, 2, 5, 7, 2, 1, 9, 1, 2, 1, 1, 2, 2, 2, 4, 2, 2, 2, 3, 8, 6, 2, 4, 3, 6, 2, 4

Offset: 1

Omar E. Pol and Robert G. Wilson v, Apr 01 2008

Also, initial digit of n-th superperfect number A019279(n), if there are no odd superperfect numbers.

			a(5)=4 because the 5th even superperfect number A061652(5) is 4096 and the initial digit of 4096 is 4.

Cf. A000030, A077648, A019279, A061652, A135613, A135617, A138125.

lst = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917}; f[n_] := Block[{pn = 2^(n - 1)}, Quotient[pn, 10^Floor[Log[10, pn]]]]; f@# & /@ (* Robert G. Wilson v, Apr 01 2008 *)

a(13)-a(39) from Robert G. Wilson v, Apr 01 2008

cp's OEIS Frontend

A138817 Concatenation of final digit of n-th Mersenne prime A000668(n), final digit of n-th even superperfect number A061652(n) and final digit of n-th perfect number A000396(n).

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A038843 Unitary superperfect numbers: numbers n such that usigma(usigma(n)) = 2*n, where usigma(n) is the sum of unitary divisors of n (A034448).

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A138842 Concatenation of initial and final digits of n-th even superperfect number A061652(n).

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A256438 Numbers m such that sigma(sigma(m-1)) = 2*(m-1).

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A272930 a(n) is the least k such that sigma(sigma(k)) = n*k, where sigma(n) is the sum of the divisors of n, or 0 if no such k exists.

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A138125 Final digit of n-th even superperfect number A061652(n).

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A138816 Concatenation of initial digit of n-th Mersenne prime A000668(n), initial digit of n-th even superperfect number A061652(n) and initial digit of n-th perfect number A000396(n).

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A138818 Concatenation of initial digit of n-th even superperfect number A061652(n), initial digit of n-th Mersenne prime A000668(n) and initial digit of n-th perfect number A000396(n).

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A138819 Concatenation of final digit of n-th even superperfect number A061652(n), final digit of n-th Mersenne prime A000668(n) and final digit of n-th perfect number A000396(n).

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